Geometric Proof of Sum Formula for Geometric Sequences
2020-03-14 08:53
This GeoGebra applet visually demonstrates the sum formula for the first n terms of a geometric sequence using geometric shapes and vector transformations. Users can adjust the first term, common ratio, and number of terms to observe the geometric representation and cumulative effect, aiding in understanding the mathematical essence of the formula.
In collections
Secondary Vocational Mathematics
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.574.0" app="classic" platform="w" id="6432756b-763d-4429-9dee-93b98b3fd871" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="765" height="545" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.3215686274509804" orientation="1" />
</panes>
<views>
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="273" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="765" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="246" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" window="50,50,500,500" />
</views>
<toolbar show="false" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="765" height="545"/>
<coordSystem xZero="215.48995996173784" yZero="182.01092555675777" scale="6.251775798117649" yscale="25.007103192470595"/>
<evSettings axes="false" grid="true" gridIsBold="false" pointCapturing="2" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="238" g="238" b="238"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="2" showNumbers="true" tickDistance="1" tickExpression="1"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="2" showNumbers="true" tickDistance="1" tickExpression="1"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="ss">
<value val="1"/>
<show object="true" label="true"/>
<condition showObject="gc ≟ 2"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="1" max="q" absoluteScreenLocation="true" width="200" x="335" y="213" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="a_1">
<value val="2"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-3" y="33"/>
<labelMode val="1"/>
<slider min="1" max="10" absoluteScreenLocation="true" width="200" x="85" y="32" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Sequence">
<input a0="Polygon[(0, 0), (a_1, 0), (a_1, -1), (0, -1)]" a1="k" a2="1" a3="1"/>
<output a0="l1a1"/>
</command>
<element type="list" label="l1a1">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<element type="numeric" label="sf">
<value val="4"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="2" max="30" absoluteScreenLocation="true" width="200" x="85" y="132" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<ggbscript onUpdate="SetAxesRatio(sf,1)
"/>
</element>
<command name="Point">
<input a0="Segment[((-sf), -1), ((-sf), -20)]"/>
<output a0="dN"/>
</command>
<element type="point" label="dN">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-4" y="-6" z="1"/>
<pointSize val="7"/>
<pointStyle val="4"/>
</element>
<expression label="nn" exp="(-y(dN))" />
<element type="numeric" label="nn">
<value val="6"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<element type="numeric" label="q">
<value val="2"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="1" max="5" absoluteScreenLocation="true" width="200" x="85" y="82" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Point">
<input a0="Segment[((-7 * sf), -1), ((-7 * sf), (-nn) - 2)]"/>
<output a0="dw"/>
</command>
<element type="point" label="dw">
<show object="true" label="true" ev="4"/>
<condition showObject="gc ≥ 4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-28" y="-8" z="1"/>
<pointSize val="5"/>
<pointStyle val="8"/>
</element>
<command name="Sequence">
<input a0="Polygon[(0, (-k) + 1), ((a_1 * q^(k - 1)), (-k) + 1), ((a_1 * q^(k - 1)), (-k)), (0, (-k))]" a1="k" a2="2" a3="nn"/>
<output a0="l1a2n"/>
</command>
<element type="list" label="l1a2n">
<show object="true" label="true"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Polygon[(0, (-k) + y(dw) + 2), ((a_1 * q^(k - 1)), (-k) + y(dw) + 2), ((a_1 * q^(k - 1)), (-k) + y(dw) + 1), (0, (-k) + y(dw) + 1)]" a1="k" a2="2" a3="nn"/>
<output a0="l1"/>
</command>
<element type="list" label="l1">
<show object="true" label="true" ev="4"/>
<condition showObject="gc ≥ 4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Polygon[(0, (-k) + y(dw) + 2), ((a_1 * q^(k - 1)), (-k) + y(dw) + 2), ((a_1 * q^(k - 1)), (-k) + y(dw) + 1), (0, (-k) + y(dw) + 1)]" a1="k" a2="nn + 1" a3="nn + 1"/>
<output a0="l2"/>
</command>
<element type="list" label="l2">
<show object="true" label="true" ev="4"/>
<condition showObject="gc ≥ 4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Text[("a_" * k) = LaTeX[(a_1 * q^(k - 1)), true, true], ((-4 * sf), (-k) + y(dw) + 2), true, true]" a1="k" a2="2" a3="nn + 1"/>
<output a0="l4"/>
</command>
<element type="list" label="l4">
<show object="true" label="true" ev="4"/>
<condition showObject="gc ≥ 4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Text[("a_" * k) = LaTeX[(a_1 * q^(k - 1)), true, true], ((-4 * sf), (-k) + 1), true, true]" a1="k" a2="1" a3="nn"/>
<output a0="l3"/>
</command>
<element type="list" label="l3">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Text">
<input a0=""\{a_n\}"" a1="((-6 * sf), ((-nn)) / 2)" a2="true" a3="true"/>
<output a0="text1"/>
</command>
<element type="text" label="text1">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint x="-24" y="-3" z="1"/>
</element>
<expression label="text5" exp=""第1行" + (LaTeX[a_1]) + "个长方形,从第2行起,每一行长方\\
形的个数是它上一行的" + (LaTeX[q]) + " 倍,共 " + (LaTeX[nn]) + "行,问一共\\
有多少个长方形?"" />
<element type="text" label="text5">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<absoluteScreenLocation x="370" y="15"/>
</element>
<command name="Text">
<input a0="(("共S_" * nn) * "个")" a1="((a_1 * q^(nn - 1) / 2), ((-nn)) / 2)" a2="true" a3="true"/>
<output a0="text3"/>
</command>
<element type="text" label="text3">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint x="32" y="-3" z="1"/>
</element>
<command name="Vector">
<input a0="((-sf), 0)" a1="dN"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="-6" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint x="-4" y="0" z="1"/>
</element>
<command name="Stretch">
<input a0="l1a1" a1="Vector[(ss, 0)]"/>
<output a0="l1a1'"/>
</command>
<element type="list" label="l1a1'">
<show object="true" label="true" ev="4"/>
<condition showObject="gc ≟ 2"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Vector">
<input a0="dN" a1="((-sf), 0)"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="6" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="dN"/>
</element>
<command name="Stretch">
<input a0="l1a2n" a1="Vector[(ss, 0)]"/>
<output a0="l1a2n'"/>
</command>
<element type="list" label="l1a2n'">
<show object="true" label="true" ev="4"/>
<condition showObject="gc ≟ 2"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<element type="numeric" label="gc">
<value val="1"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="1" max="80" absoluteScreenLocation="true" width="200" x="335" y="162" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Point">
<input a0="Segment[(((-sf)) / 2, 0), (((-sf)) / 2, (-nn) - 2)]"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<condition showObject="gc ≟ 3"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-2" y="0" z="1"/>
<pointSize val="6"/>
<pointStyle val="8"/>
</element>
<command name="Translate">
<input a0="l1a1'" a1="Vector[(0, y(A))]"/>
<output a0="l1a1''"/>
</command>
<element type="list" label="l1a1''">
<show object="true" label="true" ev="4"/>
<condition showObject="gc ≟ 3"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Translate">
<input a0="l1a2n'" a1="Vector[(0, y(A))]"/>
<output a0="l1a2n''"/>
</command>
<element type="list" label="l1a2n''">
<show object="true" label="true" ev="4"/>
<condition showObject="gc ≟ 3"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<element type="button" label="button1">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="531" y="132"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue(gc,gc-1)"/>
<font serif="false" sizeM="2" size="12" style="0"/>
<caption val="上一步"/>
</element>
<element type="button" label="button2">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="651" y="132"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue(gc,gc+1)"/>
<font serif="false" sizeM="2" size="12" style="0"/>
<caption val="下一步"/>
</element>
<command name="Text">
<input a0="(("共q·S_" * nn) * "个")" a1="((a_1 * q^(nn) / 2), y(dw) - nn / 2 + 1)" a2="true" a3="true"/>
<output a0="text4"/>
</command>
<element type="text" label="text4">
<show object="true" label="true" ev="8"/>
<condition showObject="gc ≥ 4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint x="64" y="-10" z="1"/>
</element>
<command name="Text">
<input a0=""\{ q ·a_n\}"" a1="((-6.5 * sf), y(dw) + ((-nn)) / 2)" a2="true" a3="true"/>
<output a0="text2"/>
</command>
<element type="text" label="text2">
<show object="true" label="true" ev="8"/>
<condition showObject="gc ≥ 4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint x="-26" y="-11" z="1"/>
</element>
<command name="Text">
<input a0="(((("S_" * nn) * "(q-1)=a_") * nn) * "·q-a_1")" a1="(sf, (-nn) - 4)" a2="true" a3="true"/>
<output a0="text6"/>
</command>
<element type="text" label="text6">
<show object="true" label="true" ev="8"/>
<condition showObject="gc ≥ 6"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="4" y="-10" z="1"/>
</element>
<command name="Text">
<input a0="(((("S_" * nn) * "=\frac{a_1-a_") * nn) * "·q}{1-q}")" a1="((3.5 * sf), (-nn) - 6)" a2="true" a3="true"/>
<output a0="text7"/>
</command>
<element type="text" label="text7">
<show object="true" label="true" ev="8"/>
<condition showObject="gc ≥ 7"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="14" y="-12" z="1"/>
</element>
<command name="Text">
<input a0="(((((("q×S_" * nn) * "-S_") * nn) * "=q×a_") * nn) * "-a_1")" a1="((0.2 * sf), (-nn) - 2)" a2="true" a3="true"/>
<output a0="text9"/>
</command>
<element type="text" label="text9">
<show object="true" label="true" ev="8"/>
<condition showObject="gc ≥ 5"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="0.8" y="-8" z="1"/>
</element>
<element type="button" label="button3">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="441" y="132"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue(gc,1)
SetValue(ss,1)
SetCoords(A,-sf/2,0)
SetCoords(dw,-7sf,-nn-2)"/>
<font serif="false" sizeM="2" size="12" style="0"/>
<caption val="重置"/>
</element>
</construction>
</geogebra>